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A334067 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a term of the sequence if and only if a(n) is prime" where indices start from 0.

%I A334067 #34 Dec 17 2024 10:31:52
%S A334067 1,2,3,5,6,7,11,13,14,15,16,17,18,19,23,29,31,37,41,43,44,45,46,47,48,
%T A334067 49,50,51,52,53,54,59,60,62,63,64,65,67,68,69,70,71,72,73,79,83,89,97,
%U A334067 101,103,107,109,113,127,131,132,133,134,135,137,139,140,149
%N A334067 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a term of the sequence if and only if a(n) is prime" where indices start from 0.
%C A334067 a(n) is the minimal sequence for which the sequence generated by the indices of primes in this sequence is equal to itself, where indices start from 0.
%C A334067 So if f is a function on 0-indexed integer sequences with infinitely many primes where f returns the increasing sequence of indices of primes of the input sequence b(n), then a(n) is the lexicographically minimal fixed point of f.
%C A334067 a(n) has almost the same definition as A079254, except that a(n) starts indices from 0 instead of 1. But the resulting sequences do not seem to have any correlation.
%e A334067 a(0) cannot be 0, since then 0 should be prime, which it is not.
%e A334067 a(0) = 1 is valid hence a(1) must be the next prime, which is a(1) = 2.
%e A334067 Then a(2) should be the next prime, hence a(2) = 3.
%e A334067 a(3) should be prime, hence a(3) = 5.
%e A334067 Since 4 is not in the sequence so far, a(4) must be the next nonprime, which means a(4) = 6.
%o A334067 (Python)
%o A334067 # is_prime(n) is a Python function which returns True if n is prime, and returns False otherwise. In the form stated below runs with SageMath.
%o A334067 def a_list(length):
%o A334067     """Returns the list [a(0), ..., a(length-1)]."""
%o A334067     num = 1
%o A334067     b = [1]
%o A334067     for i in range(1, length):
%o A334067         num += 1
%o A334067         if i in b:
%o A334067             while not is_prime(num):
%o A334067                 num += 1
%o A334067             b.append(num)
%o A334067         else:
%o A334067             while is_prime(num):
%o A334067                 num += 1
%o A334067             b.append(num)
%o A334067     return b
%o A334067 print(a_list(63))
%Y A334067 The same definition as A079254 except here the indices start from 0 instead of 1.
%Y A334067 Cf. A079000, A079313.
%K A334067 nonn
%O A334067 0,2
%A A334067 _Adnan Baysal_, Apr 13 2020